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On the Equivalence between Neyman Orthogonality and Pathwise Differentiability

Yuxi Chen, Edward H. Kennedy, Sivaraman Balakrishnan

Abstract

It has been frequently observed that Neyman orthogonality, the central device underlying double/debiased machine learning (Chernozhukov et al., 2018), and pathwise differentiability, a cornerstone concept from semiparametric theory, often lead to the same debiased estimators in practice. Despite the widespread adoption of both ideas, the precise nature of this equivalence has remained elusive, with the two concepts having been developed in largely separate traditions. In this work, we revisit the semiparametric framework of van der Laan and Robins (2003) and identify an implicit regularity assumption on the relationship between target and nuisance parameters -- a local product structure -- that allows us to establish a formal equivalence between Neyman orthogonality and pathwise differentiability. We demonstrate that the two directions of this equivalence impose fundamentally different structural requirements, and illustrate the theory through a concrete example of estimating the average treatment effect. This helps clarify the relationship between these two foundational frameworks and provides a useful reference for practitioners working at their intersection.

On the Equivalence between Neyman Orthogonality and Pathwise Differentiability

Abstract

It has been frequently observed that Neyman orthogonality, the central device underlying double/debiased machine learning (Chernozhukov et al., 2018), and pathwise differentiability, a cornerstone concept from semiparametric theory, often lead to the same debiased estimators in practice. Despite the widespread adoption of both ideas, the precise nature of this equivalence has remained elusive, with the two concepts having been developed in largely separate traditions. In this work, we revisit the semiparametric framework of van der Laan and Robins (2003) and identify an implicit regularity assumption on the relationship between target and nuisance parameters -- a local product structure -- that allows us to establish a formal equivalence between Neyman orthogonality and pathwise differentiability. We demonstrate that the two directions of this equivalence impose fundamentally different structural requirements, and illustrate the theory through a concrete example of estimating the average treatment effect. This helps clarify the relationship between these two foundational frameworks and provides a useful reference for practitioners working at their intersection.
Paper Structure (32 sections, 12 theorems, 162 equations)

This paper contains 32 sections, 12 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Regular Submodels and Scores
  4. Differentiating Expectations along Regular Submodels
  5. Nuisance Scores, Nuisance Tangent Space, and Pathwise Derivatives
  6. Estimating Functions and Neyman Orthogonality
  7. Local Product Structure
  8. Neyman Orthogonality
  9. The $L_2$ Chain Rule along Coordinate Paths
  10. Equivalence Between Neyman Orthogonality and Pathwise Differentiability
  11. Neyman Orthogonality Implies Pathwise Differentiability
  12. Pathwise Differentiability Implies Neyman Orthogonality
  13. Discussion
  14. Proofs of Lemmas
  15. Proof of Lemma \ref{['lem:linear_tilt_submodel']}
  16. ...and 17 more sections

Key Result

lemma 1

Let $g \in L_\infty (P_0)$ with $\bE_0[g] = 0$. Let $M := \|g\|_\infty$. For $|t| < 1/M$, define Then $p_t \ge 0$$\nu$-a.e., $\int p_t \, d\nu = 1$, and the resulting submodel $\{P_t: |t| < 1/M\}$ is regular (QMD) at $0$ with score $s \equiv g$. (Proof in Appendix appendix:proof_linear_tilt_submodel.)

Theorems & Definitions (36)

  • definition 1: Regular (QMD) submodel and score
  • definition 2: Tangent space
  • lemma 1: Linear tilt submodel is QMD with score $g$
  • corollary 1: Saturation in the nonparametric model
  • proof
  • lemma 2: Differentiation of expectations for a fixed $f$
  • lemma 3: Differentiation of expectations for varying $f_t$
  • definition 3: Nuisance scores and nuisance tangent space
  • definition 4: Pathwise differentiability and influence functions
  • remark 1: Uniqueness of the influence function
  • ...and 26 more